Let $K$ be a number field. If $X$ is a curve over $K$ with good reduction at a place $v$ of $K$, then the Jacobian of $X$ also has good reduction at $v$. This follows from the functoriality of the Jacobian.

The converse is not true, but I don't know of any examples.

Can one provide an example for all number fields $K$?

If not, I would also be pleased with just a counterexample for some place $v$ of some number field $K$